﻿ 一类水体富营养化复杂动力系统构建及其动力学分析

# 一类水体富营养化复杂动力系统构建及其动力学分析Dynamics Analysis and Construction of a Water Eutrophication Complex Dynamical System

Abstract: In this paper, firstly, on the basis of research progress of water eutrophication in Wujiayuan res-ervoir, a distributed function is introduced to characterize the total phosphorus input being changed with the season in the process of dynamic modeling, a water eutrophication complex dy-namical system has been structured. Secondly, some theoretical analysis and numerical simulation on the complex dynamical system have been investigated to establish some judgment criterions for the asymptotic stability of the internal equilibrium point and describe the dynamic trends of total phosphorus concentration and algal population density, which can dissect the interaction mechanism between total phosphorus and algal population, and then reveal the influence mecha-nism of total phosphorus input control strategy on eutrophication status and algal bloom. Finally, these studies can provide certain theoretical basis for the further prediction of the nutrient dy-namic evolution trend and the deep exploration of the algae population growth dynamic law in Wujiayuan reservoir.

1. 引言

2. 动态建模

$\left\{\begin{array}{l}\frac{\text{d}x\left(t\right)}{\text{d}t}={I}_{i}-bx\left(t\right)-\frac{ax\left(t\right)y\left(t\right)}{c+x\left(t\right)}\\ \frac{\text{d}y\left(t\right)}{\text{d}t}=ry\left(t\right)+\frac{aex\left(t\right)y\left(t\right)}{c+x\left(t\right)}-my\left(t\right)\end{array}$ (1)

3. 理论分析

$\left\{\begin{array}{l}{I}_{i}-bx\left(t\right)-\frac{ax\left(t\right)y\left(t\right)}{c+x\left(t\right)}=0\\ r+\frac{aex\left(t\right)}{c+x\left(t\right)}-m=0\end{array}$ (2)

$u\left(t\right)=x\left(t\right)-{x}_{1},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }v\left(t\right)=y\left(t\right)-{y}_{1}$ (3)

$\left\{\begin{array}{l}\frac{\text{d}u\left(t\right)}{\text{d}t}={a}_{11}u\left(t\right)+{a}_{12}v\left(t\right)\\ \frac{\text{d}v\left(t\right)}{\text{d}t}={a}_{21}u\left(t\right)+{a}_{22}v\left(t\right)\end{array}$ (4)

${a}_{11}=-b-\frac{{I}_{i}{\left(ae+r-m\right)}^{2}-bc\left(m-r\right)\left(ae+r-m\right)}{aec\left(m-r\right)}$ , ${a}_{12}=\frac{r-m}{e}$

${a}_{21}=\frac{{I}_{i}{\left(ae+r-m\right)}^{2}-bc\left(ae+r-m\right)\left(m-r\right)}{ac\left(m-r\right)}$ , ${a}_{22}=0$

4. 数值模拟与分析

5. 结论

Figure 1. The value distribution trend of key parameters

Figure 2. The simulated dynamic evolution trend of total phosphorus concentration and algal population density of the system (1)

Figure 3. The simulated dynamic evolution trend of total phosphorus concentration and algal population density of the system (1) under the condition of controlled total phosphorus

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